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Integrals

Question

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$Evaluate space colon space integral subscript negative 2 end subscript superscript 2 fraction numerator straight x squared over denominator 1 plus 5 to the power of straight x end fraction dx.$

Solution

Consider space the space given space integral straight I equals integral subscript negative 2 end subscript superscript 2 fraction numerator straight x squared over denominator 1 plus 5 to the power of straight x end fraction dx space space.... left parenthesis straight i right parenthesis Let space us space use space the space property comma integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals integral subscript straight a superscript straight b straight f left parenthesis straight a plus straight b minus straight x right parenthesis dx therefore space straight I space equals integral subscript negative 2 end subscript superscript 2 fraction numerator left parenthesis negative straight x right parenthesis squared over denominator 1 plus 5 to the power of negative straight x end exponent end fraction dx equals integral subscript negative 2 end subscript superscript 2 fraction numerator 5 to the power of straight x space straight x squared over denominator 1 plus 5 to the power of straight x end fraction dx space.. space left parenthesis ii right parenthesis Adding space equations space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space have comma 2 straight I space equals integral subscript negative 2 end subscript superscript 2 fraction numerator 1 plus 5 to the power of straight x over denominator 1 plus 5 to the power of straight x end fraction space. space straight x squared dx equals integral subscript negative 2 end subscript superscript 2 straight x squared dx equals space open square brackets straight x cubed over 3 close square brackets subscript negative 2 end subscript superscript 2 equals space 1 third left square bracket 8 minus left parenthesis negative 8 right parenthesis right square bracket equals space 1 third space left square bracket 16 right square bracket rightwards double arrow space straight I space equals space 8 over 3

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